Optimal. Leaf size=49 \[ -\frac {1}{10 a x^{10}}+\frac {b}{5 a^2 x^5}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log \left (a+b x^5\right )}{5 a^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 46}
\begin {gather*} -\frac {b^2 \log \left (a+b x^5\right )}{5 a^3}+\frac {b^2 \log (x)}{a^3}+\frac {b}{5 a^2 x^5}-\frac {1}{10 a x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^{11} \left (a+b x^5\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)} \, dx,x,x^5\right )\\ &=\frac {1}{5} \text {Subst}\left (\int \left (\frac {1}{a x^3}-\frac {b}{a^2 x^2}+\frac {b^2}{a^3 x}-\frac {b^3}{a^3 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac {1}{10 a x^{10}}+\frac {b}{5 a^2 x^5}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log \left (a+b x^5\right )}{5 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.00 \begin {gather*} -\frac {1}{10 a x^{10}}+\frac {b}{5 a^2 x^5}+\frac {b^2 \log (x)}{a^3}-\frac {b^2 \log \left (a+b x^5\right )}{5 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 44, normalized size = 0.90
method | result | size |
default | \(-\frac {1}{10 a \,x^{10}}+\frac {b}{5 a^{2} x^{5}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{5}+a \right )}{5 a^{3}}\) | \(44\) |
norman | \(\frac {-\frac {1}{10 a}+\frac {b \,x^{5}}{5 a^{2}}}{x^{10}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{5}+a \right )}{5 a^{3}}\) | \(46\) |
risch | \(\frac {-\frac {1}{10 a}+\frac {b \,x^{5}}{5 a^{2}}}{x^{10}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}-\frac {b^{2} \ln \left (b \,x^{5}+a \right )}{5 a^{3}}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 0.96 \begin {gather*} -\frac {b^{2} \log \left (b x^{5} + a\right )}{5 \, a^{3}} + \frac {b^{2} \log \left (x^{5}\right )}{5 \, a^{3}} + \frac {2 \, b x^{5} - a}{10 \, a^{2} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 45, normalized size = 0.92 \begin {gather*} -\frac {2 \, b^{2} x^{10} \log \left (b x^{5} + a\right ) - 10 \, b^{2} x^{10} \log \left (x\right ) - 2 \, a b x^{5} + a^{2}}{10 \, a^{3} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 42, normalized size = 0.86 \begin {gather*} \frac {- a + 2 b x^{5}}{10 a^{2} x^{10}} + \frac {b^{2} \log {\left (x \right )}}{a^{3}} - \frac {b^{2} \log {\left (\frac {a}{b} + x^{5} \right )}}{5 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.87, size = 55, normalized size = 1.12 \begin {gather*} -\frac {b^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{3}} + \frac {b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {3 \, b^{2} x^{10} - 2 \, a b x^{5} + a^{2}}{10 \, a^{3} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 46, normalized size = 0.94 \begin {gather*} \frac {b^2\,\ln \left (x\right )}{a^3}-\frac {b^2\,\ln \left (b\,x^5+a\right )}{5\,a^3}-\frac {\frac {1}{10\,a}-\frac {b\,x^5}{5\,a^2}}{x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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